Modelling the Induced Oscillations of Elements on a Silicon Transducer

Modelling the Induced Oscillations of Elements on a Silicon Transducer

Schoehuijs, T.

van Gijzen, M.B. (mentor)
Verweij, M.D. (mentor)

Electrical Engineering, Mathematics and Computer Science

Delft Institute of Applied Mathematics

2016-06-01

In this report, we will try to model a silicon chip with elements, as used in current experiments, with the aim to find a suitable and fast model for the transducer used. The specific question we will try to address is: How well can we model the effects of induced oscillation on other elements on a silicon chip? Induced oscillations can create problems when miniaturizing heart imaging technology for example, as the incoming signal cannot be properly converted to an image. The biharmonic equation dictates the displacement of the elements with respect to a force acting on the chip. Hence, using assumptions and numerical methods, the biharmonic equation can only be split into two coupled equations. We will look at two different boundary conditions for both of these equations; either the glue used to attach the transducer to the substrate is simply supporting the transducer, or clamping the transducer in place. We will model the biharmonic equation using Peisker's method, as well as the simply supported boundary problem. Both of these will require us to reformulate the biharmonic equation in the frequency domain, and linearize the impedance for a range between 1MHz and 9:5MHz. The results from this will then be compared to the Newmark-Beta method solution, by looking at the stability, efficiency, and accuracy. From this we can also answer the second question we will try to address: How well can Peisker's method be used to solve a fourth order differential equation? We expect that the solution to the simply supported boundary conditions look similar to the Newmark-Beta method, as they both have the same boundary conditions. Furthermore, the solution from the Peisker method should not look very different either, as they are all describing the same biharmonic equation. We found that both the Peisker method and the solution to the simply supported boundary problem look very similar, and both look like the Newmark-Beta method solution. Both methods and the non-iterative solution from the simply supported boundary problem show induced oscillations on the transducer. From this we can conclude that the simply supported boundary problem is preferred over the Newmark-Beta method due to its computation time and preferred over the Peisker method due to its computation time and smaller truncation error. The simply supported boundary problem can be more efficiently implemented than the Peisker model, as there is no need for iterations. Hence, it can be relatively quickly simulated. From this report we have seen that we can effectively model the effects of induced oscillation on other elements on a silicon chip, through the Peisker method, the Newmark-Beta method and the simply supported boundary problem. To answer our second question, Peisker's method is an effective and accurate method to model the biharmonic equation, a fourth order differential equation. However, the simply supported boundary problem is preferred.

http://resolver.tudelft.nl/uuid:d179b383-7e65-436b-ba66-5aa99aee0b4a

Student theses

bachelor thesis

(c) 2016 Schoehuijs,T.